Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

4. Disorder—Results and Limiting Cases
α > 3 < 0 leaves an all but structureless average density of state, as expected
[127].
As a rule, specific correlation-related features, like the non-analyticities at
κ = 1, tend to be washed out by integration in higher dimensional k-space.
Thus we expect arguments on general grounds [127] to hold more reliably in
higher dimensions. Conversely, the low-dimensional behavior may escape a
bird’s-view approach and require detailed calculations. We have presented
such a calculation for spatially correlated speckle disorder, so that our results
should be of immediate use for cold-atom experiments.
The non-analyticities at κ = 1 are particular features of speckle disorder.
The limiting values (4.23) and the asymptotics, however, are generic and
hold also for other models of disorder, like the Gaussian model [122]. In
figure 4.5(b), the correction of the density of states due to such a Gaussian
disorder is shown. The sharp features of the speckle disorder are washed out,
but limiting values, asymptotics and even the structure of an intermediate
minimum and maximum in one dimension are the same.
4.2. Hydrodynamic limit II: towards δ-disorder
The low-energy regime is defined by excitation energies ɛk
much smaller than the chemical potential µ. In terms of
length scales, this is phrased as ξ/λ ∝ kξ ≪ 1. In the previous
section, this has been achieved by setting the healing
length ξ to zero. For the third length scale of the system,
σ
σ
= 0 ξ
k = 0
= ∞ ξ
the disorder correlation length σ, this implied ξ ≪ σ. In order to cover also
the low-energy excitations in truly uncorrelated disorder σ < ξ, we change
the point of view in this section and realize kξ ≪ 1 by setting k to zero.
This allows describing the low-energy excitations of a disordered BEC with
arbitrary ratio σ/ξ of correlation length and healing length. The price to
pay is that kσ is constrained to be small. Compared with the Bogoliubov
wave length, the bare disorder potential is uncorrelated. The effective potential,
i.e. the density profile, is smoothed to different degrees, depending
on the ratio of ξ and σ.
4.2.1. Mean free path
In the present limit k ≪ σ −1 , ξ −1 , the inverse mean free path (4.11) evaluates
exactly the same way as in the limit (4.12), where ξ ≪ σ, k −1 and kσ ≪ 1
implied kξ ≪ 1 and kσ ≪ 1. According to (4.12), the inverse mean free
path (kls) −1 scales like (V0/µ) 2 (kσ) d . Note that the scaling l −1
s ∝ k d−1 is
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